Function value at different points

In summary, the conversation discusses the determination of a function's value at any given point using a set of known values at specific points. It is not possible to determine a unique function with only a finite set of points, and the process of finding a function that fits the given data is known as interpolation or curve fitting. Different methods, such as Lagrange interpolation, can be used to find the best fitting function.
  • #1
Apashanka
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If for a field say ##f=xyz## ,it's value is defined at 10 points in the 3-D Cartesian co-ordinate system...now using these 10 values of f and the corresponding coordinates is it possible to find the value of f at any ##(x,y,z)## of choice??
 
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  • #2
Do you mean 10 points in total, or 10 points along each axis (ie. 1000 points in total)?

Either you are trying to fit a function to given values at the grid points, in which case you to get values elsewhere you must interpolate using, for example, lagrange polynomials (MATLAB solution here), or your function is defined by an expression in closed form in which case you already know its value everywhere.
 
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  • #3
pasmith said:
Do you mean 10 points in total, or 10 points along each axis (ie. 1000 points in total)?

Either you are trying to fit a function to given values at the grid points, in which case you to get values elsewhere you must interpolate using, for example, lagrange polynomials (MATLAB solution here), or your function is defined by an expression in closed form in which case you already know its value everywhere.
(10) points in total
 
  • #4
It was the second paragraph of pasmith's that was most important.
 
  • #5
It's ok if ##f(x,y,z)## is known for 5 values of ##(x,y,z)## then how will the formula of ##f(x,y,z)## at any ##x,y,z## change??for 1-D it is simple
 
  • #6
Apashanka said:
If for a field say ##f=xyz## ,it's value is defined at 10 points in the 3-D Cartesian co-ordinate system...now using these 10 values of f and the corresponding coordinates is it possible to find the value of f at any ##(x,y,z)## of choice??

The value of ##f## at 10 points does not determine a unique value of ##f## at other points. You can invent many different functions that agree with ##f## on the 10 points but disagree with each other elsewhere. The general topic of inventing a function that has specified values on a finite number of points can be studied under the topics of "fitting a function to data" or "interpolation".

In your example, a straightforward method is to use multidimensional Lagrange interpolation. I think Answer 1 to https://math.stackexchange.com/ques...omial-in-3-d-variable-of-interest-is-a-vector describes this method.

The wikipedia gives a list of multivariate interpolation methods suited to different applications. https://en.wikipedia.org/wiki/Multivariate_interpolation
 
  • #7
No. I would think about this in 1-D first. How many points does it take to determine a quadratic function ( f(x) = ax2 +bx +c )? Would those points define a unique cubic equation?
More info here: https://en.wikipedia.org/wiki/Curve_fitting
 
  • #8
DaveE said:
How many points does it take to determine a quadratic function ( f(x) = ax2 +bx +c )? Would those points define a unique cubic equation?

The OP just said "function", not "quadratic function" or "cubic function". With no information about what kind of function it is, no finite set of points can tell you the function.
 
  • #9
PeterDonis said:
The OP just said "function", not "quadratic function" or "cubic function". With no information about what kind of function it is, no finite set of points can tell you the function.
Yes, I saw that. That's why my first sentence was "No."
I was suggesting he look at a simpler problem to highlight the limitations of curve fitting. If you can't do it for simple polynomials (even the 11th order ones), it won't work in the more general case.
 
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FAQ: Function value at different points

What is a function value at a specific point?

A function value at a specific point is the output or result of a mathematical function when a given input or value is substituted into the function. It represents the y-coordinate on a graph when the x-coordinate is a specific value.

How do you find the function value at a certain point?

To find the function value at a certain point, simply substitute the given value into the function and solve for the output. For example, if the function is f(x) = 2x + 3 and the point is (2, 7), the function value at that point would be 2(2) + 3 = 7.

Why is it important to know the function value at different points?

Knowing the function value at different points allows us to plot the points on a graph and create a visual representation of the function. This can help us understand the behavior of the function and make predictions about its values at other points.

Can the function value be the same at different points?

Yes, it is possible for a function to have the same value at different points. For example, the function f(x) = 2x will have the same value of 4 at both x = 2 and x = 4.

What does the function value represent in a real-world context?

The function value can represent a variety of things depending on the context of the function. For example, if the function represents the distance traveled over time, the function value at a certain point could represent the distance traveled at a specific time. In general, the function value represents the output or result of a mathematical relationship between two variables.

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